Yes it is certainly the same family! William and Martha had a daughter, Emma Hannah who married Henry Middleton in 1871. Their first child was my great grandfather, Henry James Middleton.
I'm still trying to find out about Martha's family. I know that her father John came from Cheshire, but I hear that looking for Axons in Cheshire is like looking for a needle in a haystack. William Beton's family are fairly easy to track though. Let me know if you want info about the Betons too.